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## Equipartition theorem
In classical statistical mechanics, the The equipartition theorem makes quantitative predictions. Like the virial theorem, it gives the k is the Boltzmann constant and _{B}T is the temperature. More generally, it can be applied to any classical system in thermal equilibrium, no matter how complicated. The equipartition theorem can be used to derive the classical ideal gas law, and the Dulong–Petit law for the specific heat capacities of solids. It can also be used to predict the properties of stars, even white dwarfs and neutron stars, since it holds even when relativistic effects are considered.
Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, namely at low enough temperatures. When the thermal energy ## Additional recommended knowledge
## Basic concept and simple examples*See also: Kinetic energy and Heat capacity*
The name "equipartition" means "share and share alike". The original concept of equipartition was that the total kinetic energy of a system is shared equally among all of its independent parts, k is the Boltzmann constant. As a consequence, the heavier atoms of xenon have a lower average speed than do the lighter atoms of helium at the same temperature. Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases.
_{B}In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of ½ k to the system's heat capacity. This has many applications.
_{B}## Translational energy and ideal gases*See also: Ideal gas*
The (Newtonian) kinetic energy of a particle of mass where v and _{y}v are the cartesian components of the velocity _{z}v. Here, H is short for Hamiltonian, and used henceforth as a symbol for energy because the Hamiltonian formalism plays a central role in the most general form of the equipartition theorem.
Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute ½ kT, as in the example of noble gases above.
_{B}More generally, in an ideal gas, the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the average total energy of an ideal gas of It follows that the heat capacity of the gas is (3/2) N=(3/2)_{A}k_{B}R, where N is Avogadro's number and _{A}R is the gas constant. Since R ≈ 2 cal/(mol·K), equipartition predicts that the molar heat capacity of an ideal gas is roughly 3 cal/(mol·K). This prediction is confirmed by experiment.^{[1]}
The mean kinetic energy also allows the root mean square speed where ^{[2]}
## Rotational energy and molecular tumbling in solution*See also: Angular velocity and Rotational diffusion*
A similar example is provided by a rotating molecule with principal moments of inertia I and _{2}I. The rotational energy of such a molecule is given by
_{3}where ω, and _{2}ω are the principal components of the angular velocity. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is (3/2)_{3}k. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated._{B}T^{[3]}
The tumbling of rigid molecules — that is, the random rotations of molecules in solution — plays a key role in the relaxations observed by nuclear magnetic resonance, particularly protein NMR and residual dipolar couplings. ## Potential energy and harmonic oscillatorsEquipartition applies to potential energies as well as kinetic energies: important examples include harmonic oscillators such as a spring, which has a quadratic potential energy where the constant p^{2}/2m, where v and p = mv denote the velocity and momentum of the oscillator. Combining these terms yields the total energy^{[6]}
Equipartition therefore implies that in thermal equilibrium, the oscillator has average energy where the angular brackets denote the average of the enclosed quantity, This result is valid for any type of harmonic oscillator, such as a pendulum, a vibrating molecule or a passive electronic oscillator. Systems of such oscillators arise in many situations; by equipartition, each such oscillator receives an average total energy ^{[8]} and the Dulong–Petit law of solid molar heat capacities. The latter application was particularly significant in the history of equipartition.
## Specific heat capacity of solids*For more details on the molar specific heat of solids, see Einstein solid and Debye model.*
An important application of the equipartition theorem is to the specific heat capacity of a crystalline solid. Each atom in such a solid can oscillate in three independent directions, so the solid can be viewed as a system of 3Nk, and its heat capacity is _{B}T3Nk.
_{B}By taking R = N between the gas constant _{A}k_{B}R and the Boltzmann constant k, this provides an explanation for the Dulong–Petit law of molar heat capacities of solids, which states that the heat capacity per mole of atoms in the lattice is _{B}3R ≈ 6 cal/(mol·K).
However, this law is inaccurate at lower temperatures, due to quantum effects; it is also inconsistent with the experimentally derived third law of thermodynamics, according to which the molar heat capacity of any substance must go to zero as the temperature goes to absolute zero. Many other physical systems can be modeled as sets of coupled oscillators. The motions of such oscillators can be decomposed into normal modes, like the vibrational modes of a piano string or the resonances of an organ pipe. On the other hand, equipartition often breaks down for such systems, because there is no exchange of energy between the normal modes. In an extreme situation, the modes are independent and so their energies are independently conserved. This shows that some sort of mixing of energies, formally called ## Sedimentation of particles*See also: Sedimentation, Mason–Weaver equation, and Brewing*
Potential energies are not always quadratic in the position. However, the equipartition theorem also shows that if a degree of freedom s) to the energy, then in thermal equilibrium the average energy of that part is k.
_{B}T/sThere is a simple application of this extension to the sedimentation of particles under gravity. where ^{[12]}
## History*This article uses the non-SI unit of*cal/(mol·K)*for molar specific heat, because it offers greater accuracy for single digits.*J/(mol·K) For an approximate conversion to the corresponding SI unit of*, such values should be multiplied by 4.2*J/cal.
The equipartition of kinetic energy was proposed initially in 1843, and more correctly in 1845, by John James Waterston.
The history of the equipartition theorem is intertwined with that of molar heat capacity, both of which were studied in the 19th century. In 1819, the French physicists Pierre Louis Dulong and Alexis Thérèse Petit discovered that the molar-specific heats of solids at room temperature were almost all identical, roughly 6 cal/(mol·K). Experimental observations of the specific heat of gases also raised concerns about the validity of the equipartition theorem. The theorem predicts that the molar heat capacity of simple monatomic gases should be roughly 3 cal/(mole·K), whereas that of diatomic gases should be roughly 7 cal/(mol·K). Experiments confirmed the former prediction, A third discrepancy concerned the specific heat of metals. N is the number of electrons. Experimentally, however, electrons contribute little to the heat capacity: the molar heat capacities of many conductors and insulators are nearly the same._{e}^{[24]}
Several explanations of equipartition's failure to account for molar heat capacities were proposed. Boltzmann defended the derivation of his equipartition theorem as correct, but suggested that gases might not be in thermal equilibrium because of their interactions with the aether. ## General formulation of the equipartition theorem*See also: Generalized coordinates, Hamiltonian mechanics, Microcanonical ensemble, and Canonical ensemble*
The most general form of the equipartition theorem m and n:
Here m=n and is zero otherwise. The averaging brackets may refer either to the long time average of a single system, or, more commonly, the ensemble average over phase space. The ergodicity assumptions implicit in the theorem imply that these two averages agree, and both have been used to estimate internal energies of complex physical systems.
The general equipartition theorem holds in both the microcanonical ensemble, The general formula is equivalent to the following two. - for all
*n*. - for all
*m*≠*n*.
If a degree of freedom a in the Hamiltonian _{n}x_{n}^{2}H, then the first of these formulae implies that
which is twice the contribution that this degree of freedom makes to the average energy . Thus the equipartition theorem for systems with quadratic energies follows easily from the general formula. A similar argument, with 2 replaced by The degrees of freedom Using the equations of Hamiltonian mechanics, Formula 2 additionally states that the averages - and
are all zero for ## Relation to the virial theorem*See also: Virial theorem, Generalized coordinates, and Hamiltonian mechanics*
The general equipartition theorem is an extension of the virial theorem (proposed in 1870 where ## Applications## Ideal gas law*See also: Ideal gas and Ideal gas law*
Ideal gases provide an important application of the equipartition theorem. As well as providing the formula for the average kinetic energy per particle, the equipartition theorem can be used to derive the ideal gas law from classical mechanics. q, _{y}q) and _{z}p = (p, _{x}p, _{y}p) denote the position vector and momentum of a particle in the gas, and
_{z}F is the net force on that particle, then
where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition formula. Summing over a system of
By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure where the divergence theorem implies that where Putting these equalities together yields which immediately implies the ideal gas law for where R=N is the gas constant.
_{A}k_{B}## Diatomic gases*See also: Two-body problem, Rigid rotor, and Harmonic oscillator*
A diatomic gas can be modelled as two masses, m, joined by a spring of stiffness _{2}a, which is called the rigid rotor-harmonic oscillator approximation.^{[17]} The classical energy of this system is
where k to the heat capacity. Therefore, the heat capacity of a gas of _{B}N diatomic molecules is predicted to be 7N · ½k: the momenta _{B}p and _{1}p contribute three degrees of freedom each, and the extension _{2}q contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be (7/2)N=(7/2)_{A}k_{B}R and, thus, the predicted molar heat capacity should be roughly 7 cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5 cal/(mol·K)^{[21]} and fall to 3 cal/(mol·K) at very low temperatures.^{[22]} This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only increase the predicted specific heat, not decrease it.^{[23]} This discrepancy was a key piece of evidence showing the need for a quantum theory of matter.
## Extreme relativistic ideal gases*See also: Special relativity, White dwarf, and Neutron star*
Equipartition was used above to derive the classical ideal gas law from Newtonian mechanics. However, relativistic effects become dominant in some systems, such as white dwarfs and neutron stars, Taking the derivative of and similarly for the p components. Adding the three components together gives
_{z}where the last equality follows from the equipartition formula. Thus, the average total energy of an extreme relativistic gas is twice that of the non-relativistic case: for ## Non-ideal gases*See also: Virial expansion and Virial coefficient*
In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through conservative forces whose potential ρ=N/V is the mean density of the gas.^{[33]} It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is
The total mean potential energy of the gas is therefore , where A similar argument, ## Anharmonic oscillators*See also: Anharmonic oscillator*
An anharmonic oscillator (in contrast to a simple harmonic oscillator) is one in which the potential energy is not quadratic in the extension where Thus, the average potential energy equals k as for the quadratic harmonic oscillator (where _{B}T/2s=2).
More generally, a typical energy function of a one-dimensional system has a Taylor expansion in the extension for non-negative integers In contrast to the other examples cited here, the equipartition formula does ## Brownian motion
*See also: Brownian motion*
The equipartition theorem can be used to derive the Brownian motion of a particle from the Langevin equation. where The dot product of this equation with the position vector for Brownian motion (since the random force and the basic equation for Brownian motion can be transformed into where the last equality follows from the equipartition theorem for translational kinetic energy: The above differential equation for (with suitable initial conditions) may be solved exactly: On small time scales, with However, on long time scales, with This describes the diffusion of the particle over time. An analogous equation for the rotational diffusion of a rigid molecule can be derived in a similar way.
## Stellar physics*See also: Astrophysics and Stellar structure*
The equipartition theorem and the related virial theorem have long been used as a tool in astrophysics. The average temperature of a star can be estimated from the equipartition theorem. where where where m is the mass of one proton. Application of the equipartition theorem gives an estimate of the star's temperature
_{p}Substitution of the mass and radius of the Sun yields an estimated solar temperature of ## Star formationThe same formulae may be applied to determining the conditions for star formation in giant molecular clouds. Assuming a constant density ρ for the cloud yields a minimum mass for stellar contraction, the Jeans mass Substituting the values typically observed in such clouds ( ## Derivations## Kinetic energies and the Maxwell–Boltzmann distributionThe original formulation of the equipartition theorem states that, in any physical system in thermal equilibrium, every particle has exactly the same average kinetic energy, (3/2) ^{[43]} This may be shown using the Maxwell–Boltzmann distribution (see Figure 2), which is the probability distribution
for the speed of a particle of mass The Maxwell–Boltzmann distribution applies to any system composed of atoms, and assumes only a canonical ensemble, specifically, that the kinetic energies are distributed according to their Boltzmann factor at a temperature as stated by the equipartition theorem. ## Quadratic energies and the partition functionMore generally, the equipartition theorem states that any degree of freedom Z(β), where β=1/(k_{B}T) is the canonical inverse temperature.^{[44]} Integration over the variable x yields a factor
in the formula for as stated by the equipartition theorem. ## General proofsGeneral derivations of the equipartition theorem can be found in many statistical mechanics textbooks, both for the microcanonical ensemble To explain these derivations, the following notation is introduced. First, the phase space is described in terms of generalized position coordinates Secondly, the infinitesimal volume
of the phase space is introduced and used to define the volume Γ( In this expression, E) is defined to be the total volume of phase space where the energy is less than E:
Since where the ellipses represent the integrand. From this, it follows that Γ is proportional to ΔE where Σ(E), and the temperature T is defined by
## The canonical ensembleIn the canonical ensemble, the system is in thermal equilibrium with an infinite heat bath at temperature where x (which could be either _{k}q or _{k}p) between two limits _{k}a and b yields the equation
where x. The first term is usually zero, either because _{k}x is zero at the limits, or because the energy goes to infinity at those limits. In that case, the equipartition theorem for the canonical ensemble follows immediately
_{k}Here, the averaging symbolized by is the ensemble average taken over the canonical ensemble. ## The microcanonical ensembleIn the microcanonical ensemble, the system is isolated from the rest of the world, or at least very weakly coupled to it. qor _{k}p) and _{k}x is given by
_{n}where the last equality follows because since the first term on the right hand side of the first line is zero (it can be rewritten as an integral of Substitution of this result into the previous equation yields Since the equipartition theorem follows: Thus, we have derived the general formulation of the equipartition theorem which was so useful in the applications described above. ## Limitations
## Requirement of ergodicity*See also: Ergodicity, Chaos theory, and Kolmogorov–Arnold–Moser theorem*
The law of equipartition holds only for ergodic systems in thermal equilibrium, which implies that all states with the same energy must be equally likely to be populated. A commonly cited counter-example where energy is ## Failure due to quantum effects*See also: Ultraviolet catastrophe, History of quantum mechanics, and Identical particles*
The law of equipartition breaks down when the thermal energy ^{[7]}^{[3]} Historically, the failures of the classical equipartition theorem to explain specific heats and blackbody radiation were critical in showing the need for a new theory of matter and radiation, namely, quantum mechanics and quantum field theory.^{[9]}
To illustrate the breakdown of equipartition, consider the average energy in a single (quantum) harmonic oscillator, which was discussed above for the classical case. Its quantum energy levels are given by h is Planck's constant, ν is the fundamental frequency of the oscillator, and n is an integer. The probability of a given energy level being populated in the canonical ensemble is given by its Boltzmann factor
where Its average energy is given by Substituting the formula for At high temperatures, when the thermal energy hν between energy levels, the exponential argument βhν is much less than one and the average energy becomes k, in agreement with the equipartition theorem (Figure 10). However, at low temperatures, when _{B}Thν >> k, the average energy goes to zero — the higher-frequency energy levels are "frozen out" (Figure 10). As another example, the internal excited electronic states of a hydrogen atom do not contribute to its specific heat as a gas at room temperature, since the thermal energy _{B}Tk (roughly 0.025 eV) is much smaller than the spacing between the lowest and next higher electronic energy levels (roughly 10 eV).
_{B}TSimilar considerations apply whenever the energy level spacing is much larger than the thermal energy. For example, this reasoning was used by Albert Einstein to resolve the ultraviolet catastrophe of blackbody radiation. ^{[47]}^{[48]} However, by the reasoning above, the average energy in the higher-ω modes goes to zero as ω goes to infinity; moreover, Planck's law of black body radiation, which describes the experimental distribution of energy in the modes, follows from the same reasoning.^{[47]}
Other, more subtle quantum effects can lead to corrections to equipartition, such as identical particles and continuous symmetries. The effects of identical particles can be dominant at very high densities and low temperatures. For example, the valence electrons in a metal can have a mean kinetic energy of a few electronvolts, which would normally correspond to a temperature of tens of thousands of kelvins. Such a state, in which the density is high enough that the Pauli exclusion principle invalidates the classical approach, is called a degenerate fermion gas. Such gases are important for the structure of white dwarf and nuetron stars. At low temperatures, a fermionic analogue of the Bose–Einstein condensate (in which a large number of identical particles occupy the lowest-energy state) can form; such superfluid electrons are responsible for superconductivity. ## See also## Notes and references- ^
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## Further reading- Huang, K (1987).
*Statistical Mechanics*, 2nd ed., John Wiley and Sons, pp. 136–138.__ISBN 0-471-81518-7__.
- Khinchin, AI (1949).
*Mathematical Foundations of Statistical Mechanics (G. Gamow, translator)*. New York: Dover Publications, pp. 93–98.__ISBN 0-486-63896-0__.
- Landau, LD; Lifshitz EM (1980).
*Statistical Physics, Part 1*, 3rd ed., Pergamon Press, pp. 129–132.__ISBN 0-08-023039-3__.
- Mandl, F (1971).
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Categories: Statistical mechanics | Thermodynamics |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Equipartition_theorem". A list of authors is available in Wikipedia. |